Abstract
This article is a survey about recent developments in the area of test sets of families of linear integer programs. Test sets are finite subsets of the integer lattice that allow to improve any given feasible non-optimal point of an integer program by one element in the set. There are various possible ways of defining test sets depending on the view that one takes: theGraver test set is naturally derived from a study of the integral vectors in cones; theScarf test set (neighbors of the origin) is strongly connected to the study of lattice point free convex bodies; the so-calledreduced Gröbner basis of an integer program is obtained from a study of generators of polynomial ideals. This explains why the study of test sets connects various branches of mathematics. We introduce in this paper these three kinds of test sets and discuss relations between them. We also illustrate on various examples such as the minimum cost flow problem, the knapsack problem and the matroid optimization problem how these test sets may be interpreted combinatorially. From the viewpoint of integer programming a major interest in test sets is their relation to the augmentation problem. This is discussed here in detail. In particular, we derive a complexity result of the augmentation problem, we discuss an algorithm for solving the augmentation problem by computing the Graver test set and show that, in the special case of an integer knapsack problem with 3 coefficients, the augmentation problem can be solved in polynomial time.
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References
Ahuja RK, Magnanti TL, Orlin JB (1993) Network flows: Theory, algorithms, and applications. Prentice Hall, Englewood Cliffs NJ
Bárány I, Howe R, Scarf H (1993) The complex of maximal lattice free simplices, Proc. of the IPCO conference Erice, Italy, pp. 1–10
Becker T, Weispfenning V (1993) Gröbner bases: A computational approach to commutative algebra. Springer Verlag, New York
Buchberger B (1985) Gröbner bases: An algorithmic method in polynomial ideal theory. In: Bose NK (ed.) Multidimensional systems theory, D. Reidel Publications, pp. 184–232
Conti P, Traverso C (1991) Buchberger algorithm and integer programming. Proceedings AAECC-9 (New Orleans), Springer LNCS 539, pp. 130–139
Cook W, Fonlupt J, Schrijver A (1986) An integer analogue of Caratheodory's theorem. J. Comb. Theory (B) 40:63–70
Cornuejols G, Urbaniak R, Weismantel R, Wolsey L (1997) Decomposition of integer programs and of generating sets. LNCS 1284, Burkard R, Woeginger G (eds.), Springer, pp. 92–103
van der Corput JG (1931) Über Systeme von linear-homogenen Gleichungen und Ungleichungen. Proceedings Koninklijke Akademie van Wetenschappen te Amsterdam 34:368–371
Cox DA, Little JB, O'Shea D (1992) Ideals, varieties, and algorithms: An introduction to computational algebraic geometry and commutative algebra. Springer-Verlag, New York
Diaconis P, Graham RL, Sturmfels B (1995) Primitive partition identities. Working Paper
Edmonds J, Karp RM (1972) Theoretical improvements in algorithmic efficiency for network flow problems. Journal of the Association for Computing Machinery 19:248–264
Faigle U (1987) Matroids in combinatorial optimization. In: White N Combinatorial geometries, Cambridge University Press, Cambridge
Frank A, Tardos E (1987) An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica 7:49–65
Gabow HN (1985) Scaling algorithms for network problems. Journal of Computer and System Sciences 31:148–168
Giles FR, Pulleyblank WR (1979) Total dual integrality and integer polyhedra. Lineare Algebra Appl. 25:191–196
Gordan P (1873) Über die Auflösung linearer Gleichungen mit reellen Coefficienten. Math. Ann. 6:23–28
Graver JE (1975) On the foundations of linear and integer programming I. Mathematical Programming 8:207–226
Gritzmann P, Wills JM (1993) Lattice points. In: Gruber PM, Wills JM (eds.) Handbook of convex geometry, Volume B, North-Holland, Amsterdam, pp. 765–798
Grötschel M, Lovász L (1995) Combinatorial optimization. In: Graham R, Grötschel M, Lovász L (eds.) Handbook of combinatorics, North-Holland, Amsterdam
Grötschel M, Lovász L, Schrijver A (1988) Geometrie algorithms and combinatorial optimization. Springer-Verlag, Berlin
Hosten S, Sturmfels B (1995) GRIN: An implementation of Gröbner bases for integer programming. In: Balas E, Clausen J (eds.) Integer programming and combinatorial optimization, Lecture notes in Computer Science 920, Springer-Verlag, Berlin, pp. 267–276
Henk M, Weismantel R (1996) On Hubert bases of polyhedral cones. Konrad-Zuse-Zentrum für Informationstechnik Berlin, Preprint SC 96-12
Kannan R (1987) Minkowski's convex body theorem and integer programming. Math. Oper. Res. 12:415–440
Kannan R, Lovász L (1988) Covering minima and lattice point free convx bodies. Ann. Math. 128:577–602
Kannan R, Lovász L, Scarf H (1990) The shapes of polyhedra. Math. Oper. Res. 15:364–380
Lagarias JC (1995) Point lattices. In: Graham R, Grötschel M, Lovász L (eds.) Handbook of combinatorics, North-Holland, Amsterdam
Liu J (1991) Hubert bases with the carathéodory property. PhD. Dissertation, Cornell University
Lovász L (1989) Geometry of numbers and integer programming. In: hi M, Tanabe K (eds.) Proc. of the 13th International Symposium on Mathematical Programming, Mathematical Programming: 177–201
Moulinet C, Pottier L Gröbner bases of toric ideals: Properties, algorithms and applications. Preprint INRIA Sophia Antipolis
Pottier L (1991) Minimal solutions of linear diophantine systems: bounds and algorithms. Proceedings RTA (Como), Springer Verlag, LNCS 488
Röck H (1980) Scaling techniques for minimal cost network flows. Discrete Structures and Algorithms, Carl Hanser, München, pp. 181–191
Scarf HE (1981) Production sets with indivisibilities, Part I: Generalities. Econometrica 49:1–32
Scarf HE (1986) Neighborhood systems for production sets with indivisibilities. Econometrica 54:507–532
Schrijver A (1986) Theory of linear and integer programming. Wiley Chichester
Schulz A, Weismantel R, Ziegler G (1995) 0/1 integer programming: Optimization and augmentation are equivalent. In: Spirakis P (ed.) Proceedings of the European Symposium on Algorithms, Lecture Notes in Computer Science, Springer-Verlag
Sebö A (1990) Hubert bases, Caratheodory's Theorem and combinatorial optimization. In: Proc. of the IPCO conference, Waterloo, Canada, pp. 431–455
Sturmfels B (1996) Gröbner bases and convex polytopes. American Mathematical Society, Providence
Sturmfels B, Thomas R (1994) Variation of cost functions in integer programming. Manuscript (1994), Mathematical Programming, to appear
Sturmfels B, Weismantel R, Ziegler G (1995) Gröbner bases of lattices, corner polyhedra and integer programming. Beitr" age zur Geometrie und Algebra 36:281–298
Thomas R (1995) A geometric Buchberger algorithm for integer programming. Math. Oper. Res. 20, 4:864–884
Thomas R (1994) Gröbner basis methods for integer programming. PhD. Dissertation, Cornell University
Thomas R, Weismantel R (1995) Truncated Gröbner bases for integer programming. Applicable Algebra in Engineering, Communication and Computing 8, 4:241–257
Urbaniak R, Weismantel R, Ziegler G (1994) A variant of Buchberger's algorighm for integer programming. SIAM Journal on Discrete Mathematics 1, 10:96–108
Welsh DJA (1976) Matroid theory. Academic Press London
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Supported by a “Gerhard-Hess-Forschungsförderpreis” of the German Science Foundation (DFG).
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Weismantel, R. Test sets of integer programs. Mathematical Methods of Operations Research 47, 1–37 (1998). https://doi.org/10.1007/BF01193834
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DOI: https://doi.org/10.1007/BF01193834